The Dedekind sum $s(p,q)$ can be both positive and negative. What are the known lower/upper bounds in terms of p,q? (I would prefer something that grows not faster than q)
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2 Answers
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For a fixed $q$, the maximum is $$s(1,q)=-{1\over4}+{1\over6q}+{q\over12}$$ and the minimum is $s(q-1,q)=-s(1,q)$.
answered May 4, 2011 at 1:09
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$\begingroup$ Wow, that's great. And 1/12 is especially delightful. Thanks! Can you also give a reference, what to cite? $\endgroup$ Commented May 4, 2011 at 12:55
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$\begingroup$ This follows from a finite-Fourier-series version of Cauchy-Schwartz. The earliest reference I'm aware of is H. Rademacher, Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956) 445-463. With some work, one can obtain even better bounds (see, e.g., <a href="front.math.ucdavis.edu/math.NT/0305421">my paper with S. Robins and S. Zacks</a>). $\endgroup$ Commented May 4, 2011 at 15:26
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$\begingroup$ $s(p,q)=q^{-2}\sum af(a)$ plus terms not relevant here, where $f(a)$ is the permutation of $1,2,\dots,q-1$ induced by multiplication by $p$ and reduction mod $q$. The maximum over all permutations $f$ (not just those arising from multiplication) of $\sum af(a)$ is attained when $f(a)=a$ for all $a$, and this corresponds to multiplication by 1; the minimum, when $f$ applied to $1,2,\dots,q-1$ gives $q-1,\dots,2,1$, and this corresponds to multiplication by $q-1$. $\endgroup$ Commented May 5, 2011 at 0:20
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$\begingroup$ Is there a known bound in terms of $s(1,q)$? I'm interested in a bound of the type $|s(a,q)| < c\cdot s(1,q)$, for some constant $c$ and any $1<a<q-1$. $\endgroup$– jiyanezCommented Jul 5, 2016 at 2:42
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$\begingroup$ @jiyanez, $c=1$ will do, of course. I think for fixed $q$ large enough and $1<a<q-1$ the maximum value of $|s(a,q)|$ is $s(2,q)$, which is not hard to evaluate. $\endgroup$ Commented Jul 5, 2016 at 6:15
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See
http://www.mathkb.com/Uwe/Forum.aspx/math/38267/upper-bounds-on-Dedekind-sums
(most of the relevant stuff is due to our own @Gerry Myerson)
answered May 3, 2011 at 17:29
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$\begingroup$ Sorry, I don't see any bound there, only some discussion on properties of the function. I'm completely ignorant in NT, I didn't mean some recent state of art result. There certainly must be some classical rough bound... $\endgroup$ Commented May 4, 2011 at 0:04
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$\begingroup$ The link seems to be kaput. kb, that's Kevin Brown, now uses mathpages.com/home but I couldn't find the Dedekind sums essay there. $\endgroup$ Commented Jul 5, 2016 at 22:48
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$\begingroup$ I found a sci.math thread titled "upper bounds on Dedekind sums" at groups.google.com/forum/#!searchin/sci.math/… and maybe the kb essay is drawn from it. $\endgroup$ Commented Jul 5, 2016 at 23:00
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$\begingroup$ That link isn't working for me, but the thread can be accessed by going to groups.google.com/g/sci.math and then searching for "upper bounds on Dedekind sums" (with the quote marks). $\endgroup$ Commented Sep 26, 2021 at 4:03